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Cosmological models and stability
Lars Andersson
Albert Einstein InstitutePotsdam
AE100 Prague, June 29, 2012
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I would already have concluded my researches about
world harmony, had not Tycho’s astronomy so
shackled me that I nearly went out of my mind
Johannes Kepler
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Cosmological principles and the standard model
The perfect cosmological principle:
The universe is isotropic in space and time
The cosmological principle:
The universe is homogenous and isotropic
The standard model in cosmology:
laws of general relativitycosmological principle
observations
Universe is approximated by Friedmann model withpositive cosmological constant Λ:
Ωm0 ∼ 0.3, ΩΛ0 ∼ 0.7
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Cosmological principles and the standard model
The perfect cosmological principle:
The universe is isotropic in space and time
The cosmological principle:
The universe is homogenous and isotropic
The standard model in cosmology:
laws of general relativitycosmological principle
observations
Universe is approximated by Friedmann model withpositive cosmological constant Λ:
Ωm0 ∼ 0.3, ΩΛ0 ∼ 0.7
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Cosmological principles and the standard model
The perfect cosmological principle:
The universe is isotropic in space and time
The cosmological principle:
The universe is homogenous and isotropic
The standard model in cosmology:
laws of general relativitycosmological principle
observations
Universe is approximated by Friedmann model withpositive cosmological constant Λ:
Ωm0 ∼ 0.3, ΩΛ0 ∼ 0.7
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Cosmological principles and the standard model
The perfect cosmological principle:
The universe is isotropic in space and time
The cosmological principle:
The universe is homogenous and isotropic
The standard model in cosmology:
laws of general relativitycosmological principle
observations
Universe is approximated by Friedmann model withpositive cosmological constant Λ:
Ωm0 ∼ 0.3, ΩΛ0 ∼ 0.7
![Page 7: Cosmological models and stability - aei.mpg.delaan/talks/prague.pdf · BKL proposal – cosmic censorship) in expanding direction: What does an observer in the late universe see?](https://reader034.vdocuments.us/reader034/viewer/2022050715/5d4acff888c993ba068bc118/html5/thumbnails/7.jpg)
Cosmological principles and the standard model
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Cosmological models and inhomogeneity
weaker cosmological principles:
statistical homogeneity
no special (matter bound) observer
Inhomogeneity in cosmological models
fitting problem
backreaction
averaging
“discrete” vs. “smooth” matter distribution
effect of inhomogeneities on observations
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Cosmological models and inhomogeneity
weaker cosmological principles:
statistical homogeneity
no special (matter bound) observer
Inhomogeneity in cosmological models
fitting problem
backreaction
averaging
“discrete” vs. “smooth” matter distribution
effect of inhomogeneities on observations
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Cosmological models and inhomogeneity
Mathematical problems
(almost) EGS
stability/instability
asymptotics
at singularity (eg. BKL proposal – cosmic censorship)
in expanding direction:
What does an observer in the late universe see?
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Cosmological models and inhomogeneity
Mathematical problems
(almost) EGS
stability/instability
asymptotics
at singularity (eg. BKL proposal – cosmic censorship)
in expanding direction:
What does an observer in the late universe see?
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Cosmological models and inhomogeneity
Mathematical problems
(almost) EGS
stability/instability
asymptotics
at singularity (eg. BKL proposal – cosmic censorship)
in expanding direction:
What does an observer in the late universe see?
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Asymptotics of cosmological models
We appear not to be in an asymptotic regime...
however we may study the mathematical problem...
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Asymptotics of cosmological models
We appear not to be in an asymptotic regime...
however we may study the mathematical problem...
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Asymptotics of cosmological models
Friedmann dust:
H2
H20
= Ω0m
(a0
a
)3
+Ω0Λ +Ω0κ
(a0
a
)2
If Λ > 0, ΩΛ dominates as a ր ∞ Restrict to ΩΛ = 0:
Einstein-de Sitter (matter dominated, Ωκ = 0):
unstable within Friedmann models, slow volume growth
a ∼ t2/3
Milne (empty universe, Ωκ = 1):
stable within Friedmann models, rapid volume growth a ∼ t
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Asymptotics of cosmological models
Friedmann dust:
H2
H20
= Ω0m
(a0
a
)3
+Ω0Λ +Ω0κ
(a0
a
)2
If Λ > 0, ΩΛ dominates as a ր ∞ Restrict to ΩΛ = 0:
Ωm = 0 Ωm = 1
MF
κ = −1 κ = 0 κ = 1
Einstein-de Sitter (matter dominated, Ωκ = 0):
unstable within Friedmann models, slow volume growth
a ∼ t2/3
Milne (empty universe, Ωκ = 1):
stable within Friedmann models, rapid volume growth a ∼ t
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Asymptotics of cosmological models
Milne is the flat interior of the lightcone in Minkowski space
Cosmological time level
line element ds2 = −dt2 + t2gH3
Deformed Milne
flat spacetime not isometric to Milne
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Asymptotics of cosmological models
Milne is the flat interior of the lightcone in Minkowski space
Cosmological time level
line element ds2 = −dt2 + t2gH3
Deformed MilneCosmological time level
flat spacetime not isometric to Milne
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Asymptotics of cosmological models
a(t) ∼ t2/3 a(t) ∼ t
deformed region has slow volume growth
deformed Milne is flat and empty – but not homogenous
and isotropic
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Asymptotics of cosmological models
More general flat models, e.g. compact quotient of deformed
Milne (Mess, 1990), (LA, 2002), (Barbot, 2005), (LA, Barbot,
Beguin & Zeghib, 2012)
Neck region – slow volume growth
Hyperbolic region
asymptotically, hyperbolic (thick) regions dominate
“neck regions” (thin) become insignificant
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Asymptotics of cosmological models
More general flat models, e.g. compact quotient of deformed
Milne (Mess, 1990), (LA, 2002), (Barbot, 2005), (LA, Barbot,
Beguin & Zeghib, 2012)
Neck region – slow volume growth
Hyperbolic region
asymptotically, hyperbolic (thick) regions dominate
“neck regions” (thin) become insignificant
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Einstein flow
The Lorentzian Einstein equations define a flow on the space of
(scale free) geometries
Consider vacuum spacetimes (M,gab)
Rab = 0
with compact Cauchy surface (M,gij ,Kij)
Use logarithmic constant mean curvature (Hubble) time
T = − ln(τ/τ0)
Consider the evolution of the scale free geometry [g] = τ2g
The Lorentzian Einstein equations define a flow
T 7→ [g](T )
on the space of Riemannian metrics
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Einstein flow
The Lorentzian Einstein equations define a flow on the space of
(scale free) geometries
Consider vacuum spacetimes (M,gab)
Rab = 0
with compact Cauchy surface (M,gij ,Kij)
Use logarithmic constant mean curvature (Hubble) time
T = − ln(τ/τ0)
Consider the evolution of the scale free geometry [g] = τ2g
The Lorentzian Einstein equations define a flow
T 7→ [g](T )
on the space of Riemannian metrics
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Einstein flow
2+1 dimensional case: Einstein equations corresponds to
time dependend Hamiltonian system on Teichmüller space
(LA, Moncrief & Tromba, 1997)M
T = −∞T = ∞
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Einstein flow
General scenario (Fischer & Moncrief, 2000), (M. Anderson,
2001)
Non-collapsing case – negative Yamabe type
For T ր ∞, (M, [g]) decomposes decomposition into
hyperbolic pieces and Seyfert fibered pieces ↔ (weak)
geometrization
Einstein flow in CMC time thick/thin decomposition of M
Thick (hyperbolic) pieces have full volume growth
⇒ in the far future, the hyperbolic pieces represent most of
the volume of M
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Einstein flow
General scenario (Fischer & Moncrief, 2000), (M. Anderson,
2001)
Non-collapsing case – negative Yamabe type
For T ր ∞, (M, [g]) decomposes decomposition into
hyperbolic pieces and Seyfert fibered pieces ↔ (weak)
geometrization
Einstein flow in CMC time thick/thin decomposition of M
Thick (hyperbolic) pieces have full volume growth
⇒ in the far future, the hyperbolic pieces represent most of
the volume of M
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Einstein flow
Nonlinear stability problem for cosmological models:
Prove that for Cauchy data close to Milne, the future
Cauchy development is asymptotic to Milne
Vacuum case: (LA & Moncrief, 2004), (LA & Moncrief, 2011)
More general question:
For Cauchy data close to κ ≤ 0 Friedmann,
characterize the future Cauchy development
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Einstein flow
Nonlinear stability problem for cosmological models:
Prove that for Cauchy data close to Milne, the future
Cauchy development is asymptotic to Milne
Vacuum case: (LA & Moncrief, 2004), (LA & Moncrief, 2011)
More general question:
For Cauchy data close to κ ≤ 0 Friedmann,
characterize the future Cauchy development
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Nonlinear stability: Minkowski
(Friedrich, 1986): nonlinear stability to the
future of a hyperboloidal slice, regular I+
(Christodoulou & Klainerman, 1993;Klainerman & Nicolò, 2003): use null
coordinates, Bel-Robinson energy. Getpeeling if sufficiently regular at i0
(Lindblad & Rodnianski, 2005): use weak
null condition, simple proof, matter can beadded
I+
i0
I−
conformal type:
Minkowski diamond
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Nonlinear stability: Dark Energy
future horizons, topology does not matter
(but cf. (LA & Galloway, 2002))
Locality at I+ ⇒ “small data” can be charac-
terized locally in spaceResults:
(Friedrich, 1991),
(M. Anderson & Chrusciel,
2005) (Heinzle & Rendall,
2005): global stability
(Starobinsky, 1983),
(Rendall, 2006):
expansions, Fuchsian
(Ringström, 2007): “local”
small data global
existence
(Einstein-Λ-scalar field)
I+
conformal type:
finite cylinder
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Nonlinear stability: Dark Energy
future horizons, topology does not matter
(but cf. (LA & Galloway, 2002))
Locality at I+ ⇒ “small data” can be charac-
terized locally in spaceResults:
Einstein-Λ-irrotational fluid
(Rodnianski & Speck,
2009)
Einstein-Λ-Euler (Speck,
2011)
Einstein-Λ-Vlasov
(Ringström, 2012)
I+
conformal type:
finite cylinder
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Nonlinear stability: Ordinary matter
Example: Lorentz cone on compact
hyperbolic: ds2 = −dt2 + t2γH(κ = −1 empty Friedmann)
topology matters
vacuum:
U(1) (Choquet-Bruhat & Moncrief,
2001)
G0 (LA & Moncrief, 2004), (LA &
Moncrief, 2011)
matter:
Einstein-Vlasov, Bianchi symmetry:
(Rendall & Tod, 1999), (Heinzle &
Uggla, 2006), (Nungesser, 2011)
2+1 Einstein-Vlasov: (Fajman, 2012)
test fluids on FLRW: (Speck, 2012)
conformal type:
infinite cylinder
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Nonlinear stability: Ordinary matter
Example: Lorentz cone on compact
hyperbolic: ds2 = −dt2 + t2γH(κ = −1 empty Friedmann)
topology matters
vacuum:
U(1) (Choquet-Bruhat & Moncrief,
2001)
G0 (LA & Moncrief, 2004), (LA &
Moncrief, 2011)
matter:
Einstein-Vlasov, Bianchi symmetry:
(Rendall & Tod, 1999), (Heinzle &
Uggla, 2006), (Nungesser, 2011)
2+1 Einstein-Vlasov: (Fajman, 2012)
test fluids on FLRW: (Speck, 2012)
conformal type:
infinite cylinder
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Scale invariant variables
(M, γ) n-dimensional negative Einstein
Ricγ = −n − 1
n2γ ; ds2 = −dt2 +
t2
n2γ Lorentz cone over γ
line element ds2 = −N2dt2 + gij(dx i + X idt)(dx j + X jdt)
(physical) vacuum data: (M, g, K , N, X )
τ = g ij Kij mean curvature (assume CMC time gauge)
Rescaled fields (g,Σ,N,X ) :
gij = τ2gij , Σij = τ(Kij −τ
ngij)
X i = τ X i N = τ2N
Scale invariant time: T = − log(−τ)At background: (gij ,Σij ,X
i ,N) = (γij ,0,0,n)
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Scale invariant variables
(M, γ) n-dimensional negative Einstein
Ricγ = −n − 1
n2γ ; ds2 = −dt2 +
t2
n2γ Lorentz cone over γ
line element ds2 = −N2dt2 + gij(dx i + X idt)(dx j + X jdt)
(physical) vacuum data: (M, g, K , N, X )
τ = g ij Kij mean curvature (assume CMC time gauge)
Rescaled fields (g,Σ,N,X ) :
gij = τ2gij , Σij = τ(Kij −τ
ngij)
X i = τ X i N = τ2N
Scale invariant time: T = − log(−τ)At background: (gij ,Σij ,X
i ,N) = (γij ,0,0,n)
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Stable Riemannian Einstein spaces
Require stability of (M, γ):
L ≥ 0,
where Lhac = −∆hac − 2Rabcdhbd
Allow a nontrivial, integrable moduli space of Riemannian
Einstein structures
stability and integrability holds in all known cases for
negative Einstein spaces
n = 3: Mostow rigidity ⇒ trivial moduli space
n > 3 The moduli space of Einstein structures correspondsto
center manifold for normalized Ricci flow
center manifold for Lorentzian Einstein flow
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Stable Riemannian Einstein spaces
Require stability of (M, γ):
L ≥ 0,
where Lhac = −∆hac − 2Rabcdhbd
Allow a nontrivial, integrable moduli space of Riemannian
Einstein structures
stability and integrability holds in all known cases for
negative Einstein spaces
n = 3: Mostow rigidity ⇒ trivial moduli space
n > 3 The moduli space of Einstein structures correspondsto
center manifold for normalized Ricci flow
center manifold for Lorentzian Einstein flow
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Linearized stability analysis
Linearizing the rescaled Einstein equations around
background data (g,Σ,N,X ) = (γ,0,n,0) gives
X + (n − 1)X + n2λX = 0
damped oscillator equation with characteristic roots
−(n − 1)±√
(n − 1)2 − 4n2λ
2
Energy E = 12 X 2 + n2λ
2 X 2 + cEXX
Energy decay: ddT
E ≤ α+E
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Linearized stability analysis
Linearizing the rescaled Einstein equations around
background data (g,Σ,N,X ) = (γ,0,n,0) gives
X + (n − 1)X + n2λX = 0
damped oscillator equation with characteristic roots
−(n − 1)±√
(n − 1)2 − 4n2λ
2
Energy E = 12 X 2 + n2λ
2 X 2 + cEXX
Energy decay: ddT
E ≤ α+E
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Linearized stability analysis
C
anomalous
λ < (n−1)2
4n2
marginalλ = (n−1)2
4n2
universal
λ > (n−1)2
4n2
Let λ0 = smallest non-zero eigenvalue of L. Define
cE =
n−12
2n2λ0
n−1
α+ =
−n−12
universal
− (n−1)+√
(n−1)2−4n2λ0
2 anomalous
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Energies
Use linearized analysis as a guide to defining energies:
write energies modelled on damped harmonic oscillator
energy, in terms of variables g − γ,Σ
energy decay for small data
scale invariant geometry converges to Einstein geometry
in the moduli space:
⇒ Einstein spaces are attractors for the Einstein flow
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Energies
Use linearized analysis as a guide to defining energies:
write energies modelled on damped harmonic oscillator
energy, in terms of variables g − γ,Σ
energy decay for small data
scale invariant geometry converges to Einstein geometry
in the moduli space:
⇒ Einstein spaces are attractors for the Einstein flow
![Page 43: Cosmological models and stability - aei.mpg.delaan/talks/prague.pdf · BKL proposal – cosmic censorship) in expanding direction: What does an observer in the late universe see?](https://reader034.vdocuments.us/reader034/viewer/2022050715/5d4acff888c993ba068bc118/html5/thumbnails/43.jpg)
Energies
Use linearized analysis as a guide to defining energies:
write energies modelled on damped harmonic oscillator
energy, in terms of variables g − γ,Σ
energy decay for small data
scale invariant geometry converges to Einstein geometry
in the moduli space:
⇒ Einstein spaces are attractors for the Einstein flow
![Page 44: Cosmological models and stability - aei.mpg.delaan/talks/prague.pdf · BKL proposal – cosmic censorship) in expanding direction: What does an observer in the late universe see?](https://reader034.vdocuments.us/reader034/viewer/2022050715/5d4acff888c993ba068bc118/html5/thumbnails/44.jpg)
Energies
Use linearized analysis as a guide to defining energies:
write energies modelled on damped harmonic oscillator
energy, in terms of variables g − γ,Σ
energy decay for small data
scale invariant geometry converges to Einstein geometry
in the moduli space:
⇒ Einstein spaces are attractors for the Einstein flow
![Page 45: Cosmological models and stability - aei.mpg.delaan/talks/prague.pdf · BKL proposal – cosmic censorship) in expanding direction: What does an observer in the late universe see?](https://reader034.vdocuments.us/reader034/viewer/2022050715/5d4acff888c993ba068bc118/html5/thumbnails/45.jpg)
Stability theorem
Theorem (LA & Moncrief, 2011)
Suppose (Mn, γ0) stable, integrable, Ricγ0 = −n−1n2 γ0 and let
vacuum data (g0, K 0) be given.
Assume g0 = τ2g0,Σ0 = τ(K − τn) are close to (γ0,0),
Then, the maximal vacuum Cauchy development (M,g) of
(M, g0, K 0)
has a global CMC foliation to the future,
is future causally geodesically complete,
g(T ) → γ∞ ∈ Nγ0 , as T → ∞
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Stability theorem
Theorem (LA & Moncrief, 2011)
Suppose (Mn, γ0) stable, integrable, Ricγ0 = −n−1n2 γ0 and let
vacuum data (g0, K 0) be given.
Assume g0 = τ2g0,Σ0 = τ(K − τn) are close to (γ0,0),
Then, the maximal vacuum Cauchy development (M,g) of
(M, g0, K 0)
has a global CMC foliation to the future,
is future causally geodesically complete,
g(T ) → γ∞ ∈ Nγ0 , as T → ∞
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Generalized Kasner spacetimes
(LA & Heinzle, 2006)
Generalized Kasner spaces: M ∼= R× M × N, with
(Mm,g), (Nn,h), D = d + 1 = m + n + 1
Ricg = −(m + n − 1)g, Rich = −(m + n − 1)hline element ds2 = −dt2 + a2(t)g + b2(t)h
introduce scale invariant variables
p = −a/a, q = −b/b,
P = p/H, Q = q/H, A = 1aH , B = 1
bH
Einstein equations ⇒ autonomous system for (P,Q,A,B)with 2 constraints.
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Generalized Kasner spacetimes
(LA & Heinzle, 2006)
Generalized Kasner spaces: M ∼= R× M × N, with
(Mm,g), (Nn,h), D = d + 1 = m + n + 1
Ricg = −(m + n − 1)g, Rich = −(m + n − 1)hline element ds2 = −dt2 + a2(t)g + b2(t)h
introduce scale invariant variables
p = −a/a, q = −b/b,
P = p/H, Q = q/H, A = 1aH , B = 1
bH
Einstein equations ⇒ autonomous system for (P,Q,A,B)with 2 constraints.
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Generalized Kasner spacetimes
Dynamical systems analysis shows the generic orbit is
generalized Kasner (↔ a ∼ tp, b ∼ tq ) at singularity,
Friedmann (↔ cone) in expanding direction
Friedmann is stable node only if spacetime dimension
D ≥ 11
(F1)
(F2)
(FA)
(FB)
A=
0
B=
0
D < 10
(F1)
(F2)
(FA)
(FB)
A=
0
B=
0
D > 10
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Generalized Kasner spacetimes
Quiescent singularities:
(Demaret, Henneaux, & Spindel, 1985): condition for
quiescent behavior at singularity in D = d + 1 dimensions:
1 + p1 − pd − pd−1 > 0 (1)
where pa = generalized Kasner exponents at singularity.
Heuristic: Eq. (1) holds in vacuum only if D ≥ 11 ⇒generic vacuum, D < 11 spacetime has oscillatorysingularity,
generic vacuum, D ≥ 11 spacetime has quiescent
singularity
generic D = 4 spacetime with scalar field has quiescent
singularity (LA & Rendall, 2001), Fuchsian analysis
generic D ≥ 11 vacuum spacetime has quiescent
singularity (Damour, Henneaux, Rendall, & Weaver, 2002),
Fuchsian analysis
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Generalized Kasner spacetimes
Quiescent singularities:
(Demaret et al., 1985): condition for quiescent behavior at
singularity in D = d + 1 dimensions:
1 + p1 − pd − pd−1 > 0 (1)
where pa = generalized Kasner exponents at singularity.
Heuristic: Eq. (1) holds in vacuum only if D ≥ 11 ⇒generic vacuum, D < 11 spacetime has oscillatorysingularity,
generic vacuum, D ≥ 11 spacetime has quiescentsingularity
generic D = 4 spacetime with scalar field has quiescent
singularity (LA & Rendall, 2001), Fuchsian analysis
generic D ≥ 11 vacuum spacetime has quiescent
singularity (Damour et al., 2002), Fuchsian analysis
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From α to ω
Global nonlinear stability in the real analytic category:
Theorem (LA, 2009)
Suppose Mm,Nn stable, integrable with D = m + n + 1 ≥ 11.
Then there is a full-parameter family of Cω Cauchy data on
M × N, such that the maximal Cauchy development (M,g)
has global CMC time function,
has quiescent, crushing singularity,
is future causally complete,
is asymptotically Friedmann to the future,
g(T ) → γM∞
+ γN∞
, as T → ∞.
This applies to a large variety of factors M,N, and can easily be
generalized to multiple factors to give rich future asymptotics
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From α to ω
Global nonlinear stability in the real analytic category:
Theorem (LA, 2009)
Suppose Mm,Nn stable, integrable with D = m + n + 1 ≥ 11.
Then there is a full-parameter family of Cω Cauchy data on
M × N, such that the maximal Cauchy development (M,g)
has global CMC time function,
has quiescent, crushing singularity,
is future causally complete,
is asymptotically Friedmann to the future,
g(T ) → γM∞
+ γN∞
, as T → ∞.
This applies to a large variety of factors M,N, and can easily be
generalized to multiple factors to give rich future asymptotics
![Page 54: Cosmological models and stability - aei.mpg.delaan/talks/prague.pdf · BKL proposal – cosmic censorship) in expanding direction: What does an observer in the late universe see?](https://reader034.vdocuments.us/reader034/viewer/2022050715/5d4acff888c993ba068bc118/html5/thumbnails/54.jpg)
Concluding remarks/Open problems
Future asymptotics of cosmological models well
understood in highly symmetric cases: Friedmann,
Bianchi, Gowdy, T 2, U(1) – full 3+1 case mostly open
Prove nonlinear stability of Milne for Einstein-Vlasov
Characterize future evolution of inhomogenous
Einstein-matter spacetimes close to Friedmann. Which
cases are nonlinearly stable?
Numerical studies of cosmological models in GR beyond
LTB/spherical symmetry
![Page 55: Cosmological models and stability - aei.mpg.delaan/talks/prague.pdf · BKL proposal – cosmic censorship) in expanding direction: What does an observer in the late universe see?](https://reader034.vdocuments.us/reader034/viewer/2022050715/5d4acff888c993ba068bc118/html5/thumbnails/55.jpg)
Concluding remarks/Open problems
Future asymptotics of cosmological models well
understood in highly symmetric cases: Friedmann,
Bianchi, Gowdy, T 2, U(1) – full 3+1 case mostly open
Prove nonlinear stability of Milne for Einstein-Vlasov
Characterize future evolution of inhomogenous
Einstein-matter spacetimes close to Friedmann. Which
cases are nonlinearly stable?
Numerical studies of cosmological models in GR beyond
LTB/spherical symmetry
![Page 56: Cosmological models and stability - aei.mpg.delaan/talks/prague.pdf · BKL proposal – cosmic censorship) in expanding direction: What does an observer in the late universe see?](https://reader034.vdocuments.us/reader034/viewer/2022050715/5d4acff888c993ba068bc118/html5/thumbnails/56.jpg)
Concluding remarks/Open problems
Future asymptotics of cosmological models well
understood in highly symmetric cases: Friedmann,
Bianchi, Gowdy, T 2, U(1) – full 3+1 case mostly open
Prove nonlinear stability of Milne for Einstein-Vlasov
Characterize future evolution of inhomogenous
Einstein-matter spacetimes close to Friedmann. Which
cases are nonlinearly stable?
Numerical studies of cosmological models in GR beyond
LTB/spherical symmetry
![Page 57: Cosmological models and stability - aei.mpg.delaan/talks/prague.pdf · BKL proposal – cosmic censorship) in expanding direction: What does an observer in the late universe see?](https://reader034.vdocuments.us/reader034/viewer/2022050715/5d4acff888c993ba068bc118/html5/thumbnails/57.jpg)
Concluding remarks/Open problems
Future asymptotics of cosmological models well
understood in highly symmetric cases: Friedmann,
Bianchi, Gowdy, T 2, U(1) – full 3+1 case mostly open
Prove nonlinear stability of Milne for Einstein-Vlasov
Characterize future evolution of inhomogenous
Einstein-matter spacetimes close to Friedmann. Which
cases are nonlinearly stable?
Numerical studies of cosmological models in GR beyond
LTB/spherical symmetry
![Page 58: Cosmological models and stability - aei.mpg.delaan/talks/prague.pdf · BKL proposal – cosmic censorship) in expanding direction: What does an observer in the late universe see?](https://reader034.vdocuments.us/reader034/viewer/2022050715/5d4acff888c993ba068bc118/html5/thumbnails/58.jpg)
Concluding remarks/Open problems
Future asymptotics of cosmological models well
understood in highly symmetric cases: Friedmann,
Bianchi, Gowdy, T 2, U(1) – full 3+1 case mostly open
Prove nonlinear stability of Milne for Einstein-Vlasov
Characterize future evolution of inhomogenous
Einstein-matter spacetimes close to Friedmann. Which
cases are nonlinearly stable?
Numerical studies of cosmological models in GR beyond
LTB/spherical symmetry
Thank You
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References I
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Andersson, L. (2002). Constant mean curvature foliations of
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References II
Andersson, L., & Galloway, G. J. (2002). dS/CFT and
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References III
Andersson, L., Moncrief, V., & Tromba, A. J. (1997). On the
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Andersson, L., & Rendall, A. D. (2001). Quiescent cosmological
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Barbot, T. (2005). Globally hyperbolic flat space-times. J.
Geom. Phys., 53(2), 123–165. Retrieved from
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doi: 10.1016/j.geomphys.2004.05.002
Choquet-Bruhat, Y., & Moncrief, V. (2001). Future global in time
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Christodoulou, D., & Klainerman, S. (1993). The global
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Princeton University Press.
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References IV
Damour, T., Henneaux, M., Rendall, A. D., & Weaver, M. (2002,
February). Kasner-Like Behaviour for Subcritical
Einstein-Matter Systems. NASA STI/Recon Technical
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Demaret, J., Henneaux, M., & Spindel, P. (1985).
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Fajman, D. (2012).
Fischer, A. E., & Moncrief, V. (2000). Hamiltonian reduction of
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Edge, NJ: World Sci. Publishing.
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References V
Friedrich, H. (1986). On the existence of n-geodesically
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Friedrich, H. (1991). On the global existence and the
asymptotic behavior of solutions to the
Einstein-Maxwell-Yang-Mills equations. J. Differential
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Heinzle, J. M., & Rendall, A. D. (2005). Power-law inflation in
spacetimes without symmetry. Retrieved from
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Heinzle, J. M., & Uggla, C. (2006, May). Dynamics of the
spatially homogeneous Bianchi type I Einstein Vlasov
equations. Classical and Quantum Gravity, 23,
3463-3489. doi: 10.1088/0264-9381/23/10/016
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References VI
Klainerman, S., & Nicolò, F. (2003). Peeling properties of
asymptotically flat solutions to the Einstein vacuum
equations. Classical and Quantum Gravity, 20(14),
3215-3257.
Lindblad, H., & Rodnianski, I. (2005). Global existence for the
einstein vacuum equations in wave coordinates. Commu-
nications in Mathematical Physics, 256, 43. Retrieved from
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Mess, G. (1990). Lorentz spacetimes of constant curvature
(Tech. Rep. No. IHES/M/90/28). Institute des Hautes
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Einstein-Vlasov system with Bianchi I symmetry. Journal
of Physics Conference Series, 314(1), 012097. doi:
10.1088/1742-6596/314/1/012097
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References VII
Rendall, A. D. (2006). Mathematical properties of cosmological
models with accelerated expansion. In Analytical and
numerical approaches to mathematical relativity (Vol. 692,
pp. 141–155). Berlin: Springer.
Rendall, A. D., & Tod, K. P. (1999, June). Dynamics of spatially
homogeneous solutions of the Einstein-Vlasov equations
which are locally rotationally symmetric. Classical and
Quantum Gravity, 16, 1705-1726. doi:
10.1088/0264-9381/16/6/305
Ringström, H. (2012). On the topology and future stability of
models of the universe - with an introduction to the
Einstein-Vlasov system. under preparation.
Rodnianski, I., & Speck, J. (2009, November). The Stability of
the Irrotational Euler-Einstein System with a Positive
Cosmological Constant. ArXiv e-prints.
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References VIII
Speck, J. (2011, February). The Nonlinear Future-Stability of
the FLRW Family of Solutions to the Euler-Einstein
System with a Positive Cosmological Constant. ArXiv
e-prints.
Speck, J. (2012, January). The Stabilizing Effect of Spacetime
Expansion on Relativistic Fluids With Sharp Results for
the Radiation Equation of State. ArXiv e-prints.
Starobinsky, A. A. (1983). Isotropization of arbitrary
cosmological expansion given an effective cosmological
constant. JETP Lett., 37, 66-69.